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Chapter 4: Problem 28

Graph each equation by finding the intercepts and at least one other point. $$x=0$$

Short Answer

Expert verified
The equation \(x=0\) is a vertical line and has a y-intercept at (0, y) for any value of y. Since the line is vertical and remains at x=0, the x-intercept doesn't exist. Another point on the line is (0, 1). To graph the line, plot the y-intercept and the point (0, 1), then draw a vertical line through these points. The line will pass through every point on the y-axis but will never touch the x-axis.

Step by step solution

01

Find the y-intercept

To find the y-intercept, we set \(x=0\) (which is already given) and find the corresponding value of \(y\). In this case, it doesn't matter what value of \(y\) we have, because x will always be equal to zero. So the y-intercept is at point (0, y) for any value of y.
02

Find the x-intercept

For this graph, there is no x-intercept as the line does not touch the x-axis. The line is vertical and remains at x = 0 throughout its length.
03

Find at least one other point on the line

Since the line is vertical and remains at x = 0, we can choose any value for y to find another point. For simplicity, let's choose \(y=1\). The point will then be (0, 1).
04

Graph the line

Plot the y-intercept, the point (0, 1), and draw a vertical line through these points. This is the graph of the equation \(x = 0\). The line will pass through every point on the y-axis, and it will never touch the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Intercepts in Graphing
Intercepts are crucial in graphing linear equations, helping us find where a line crosses the x-axis or y-axis. Intercepts give us specific points that help anchor the line on the graph.

  • Y-intercept: This is the point where the line crosses the y-axis. For instance, if you set \(x = 0\), you find the y-intercept. In the case of a vertical line like \(x = 0\), the y-intercept is actually not fixed at a single point. Instead, it can be any point along the y-axis because the line covers the entire vertical span.
  • X-intercept: This is where the line crosses the x-axis. For \(x = 0\), there is no x-intercept because the line is vertical and remains on the y-axis instead.
Knowing how to find intercepts allows for quick and easy graph plotting, especially by giving key reference points for any linear equation.
Exploring Vertical Lines
Vertical lines have unique characteristics compared to other types of lines. One defining feature is that they run parallel to the y-axis.

  • No Slope: Since vertical lines go straight up and down, they don't have a slope. In mathematical terms, their slope is considered undefined.
  • Equation Form: The equation for a vertical line is always in the form \(x = \text{constant}\). For example, \(x = 0\) tells us that the line is located at \(x = 0\) and covers all possible y-values.
  • Graphing: To graph vertical lines, plot a straight line that runs parallel to the y-axis at the given x-value, covering every y-value.
Recognizing vertical lines is essential for correctly interpreting and graphing figures on the coordinate plane.
Navigating the Coordinate Plane
The coordinate plane provides a framework for plotting points and lines by using a grid system.

  • Axes: The plane is divided by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin \((0, 0)\).
  • Quadrants: There are four sections known as quadrants, numbered counterclockwise starting from the upper right. The position of a point is determined based on these quadrants.
  • Plotting Points: Every point on the plane is identified by a pair \((x, y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.
Using the coordinate plane effectively allows you to visualize linear equations and other algebraic concepts, making complex problems easier to solve.

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Most popular questions from this chapter

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(4,1),(4,3)\\\&L_{2}:(1,0),(1,-2)\end{aligned}$$

Graph each function. $$C(h)=1.5 h$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=-\frac{3}{8 x-5}$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=-\frac{1}{4} x-2$$

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(y=-x-8\) containing \((4,11)\)
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